Quincy attended Harvard College, graduating in 1708. He married Elizabeth Norton, daughter of the Reverend John Norton of Hingham, on 3 September 1715. Although officially only achieving the rank of major in the British army, he was commonly referred to as "Colonel".

In 1717, he was elected to represent Braintree at the Massachusetts General Court, was re-elected in 1719, and served in that capacity until 1740. From 1729 to 1741, he served as the Speaker of the House. In 1741, Quincy was voted out of office, but was returned there in 1744, where he served four additional years. He died on 13 July 1767 at the age of 77, only two days after the birth of his namesake, John Quincy Adams.